Inclusion-exclusion enhanced by nerve stimulation

TL;DR

This paper introduces a method to optimize inclusion-exclusion calculations by leveraging nerve structures and term collection, reducing computational effort in combinatorial evaluations.

Contribution

It presents a novel approach combining nerve-based encoding and term collection to efficiently evaluate inclusion-exclusion expansions.

Findings

Significant reduction in computational complexity for inclusion-exclusion calculations.

Effective encoding of nonzero term structures using nerve sets.

Enhanced efficiency through collection of equal nonzero terms.

Abstract

When evaluating the lengthy inclusion-exclusion expansion many of its terms may turn out to be zero, and hence should be discarded beforehand. Often this can be done. The main idea is that the index sets of nonzero terms constitute a set ideal (called the 'nerve'), which often can be encoded in a compact way (Upgrade B). As a further enhancement (Upgrade A), equal nonzero terms can sometimes be efficiently collected.